Misfit weighted

Let us introduce some weighting factors w for estimation of the residuals Ti. The reason for the weighing is that in practice some observations are made with more accuracy than others. In this case one would like the prediction errors rj of the more accurate observations to have a greater weight than the inaccurate observations. To accomplish this weighting we define the weighted misfit functional as follows... 

The problem of minimization of the weighted misfit functional can be solved by calculating the first variation of this functional and setting it equal to zero ... 

Therefore, on each iteration of the re-weighted RCG method we actually minimize the parametric functional with the different stabilizers, because the weighting matrix Wen is updated on each iteration. In order to insure the convergence of the misfit functional to the global minimum, we use adaptive regularization and decrease the ttn+i, if 7 > 1 ... 

Figure 7-5 The result of traditional minimum norm inversion without model parameter weights. The bottom panel shows the density distribution obtained after 32 iterations. The top panel presents the normalized misfit functional versus iteration number...

Figure 7-7 illustrates the focusing inversion result obtained by the re-weighted regularized conjugate gradient method. The plots of the misfit and parametric functionals are shown in the top panel of Figure 7-7. In this case the data fitting after 50 iterations is within 4% nevertheless the inverse image adec uately reconstructs the true model. We can clearly recognize two bodies in this image, and the densities correspond well to the true model.

Waldhauser and Ellsworth [2000] use a two-step iterative solution procedure, where a priori weights describing data quality are first applied to the arrival times. Once a stable solution is obtained, the data are reweighted by multiplying the a priori quality weights with values that depend on the misfit of the data from the previous iteration and on the offset between events (to downweight event pairs with large inter-event differences). 

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